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v
r
cos
θ
−
ln
(
1
−
A
(
v
)) =
g
(
d
(
v
))
dv
.
(15.6)
s
−
1
Broom and Ruxton [
2
] considered three cases, and we shall briefly look at two of
them. A summary of the important model parameters is given in Table
15.1
.Rewards
to both predator and prey from the three different possible types of interaction are
shown in Table
15.2
.
15.3.2 The Predator Attacks As Soon As the Prey Is Observed
In this version of the model, the predator must attack as soon as it detects the prey,
and can see behind itself (and so may still attack after the closest position).
The payoff for never fleeing unless attacked is given by
1
dA
(
v
)
R
(
N
)=
1
{
1
−
f
(
d
(
v
))
}
(
1
−
c
)
dv
+
1
−
A
(
1
)
.
(15.7)
dv
−
Strategy
V
is to flee if the predator attacks or if the predator reaches position
v
=
V
without attacking, for some position
V
∈
[
−
1
,
1
]
. The payoff from this strategy
is
V
dA
(
v
)
R
(
V
)=
1
{
1
−
f
(
d
(
v
))
}
(
1
−
c
)
dv
+(
1
−
A
(
V
))(
1
−
c
)
{
1
−
f
(
d
(
V
)+
Δ
)
},
dv
−
(15.8)
where
is the extra distance that the prey can move when it initiates a chase before
the predator is aware of the fleeing prey (so that effectively the chase starts with a
distance
v
Δ
between the two individuals, see Fig.
15.1
c).
The case of fleeing immediately is simply
V
+
Δ
=
−
1intheabovei.e.
R
(
−
1
)=(
1
−
f
(
r
+
Δ
))(
1
−
c
)
.
(15.9)
> −
(
)
(
−
)(
−
For any
V
1,
R
V
is a weighted average of terms of the form
1
c
1
(
))
(
−
)
(
−
)
>
(
)
f
d
that are never bigger than
R
1
,andso
R
1
R
V
.
Situation
Prey's payoff
Predator's payoff
No chase
1
0
Attack-initiated chase
(
1
−
c
)(
1
−
f
(
d
(
v
)))
f
(
d
(
v
))
Flight-initiated chase
(
1
−
c
)(
1
−
f
(
d
(
v
)+
Δ
))
f
(
d
(
v
)+
Δ
)
Table 15.2
The possible interactions and associated payoffs for the model of Broom and
Ruxton [
2
]
Thus the optimal strategy is either to run immediately (with payoff
R
(
−
1
)
)or
only to run when the predator initiates an attack (with payoff
R
(
N
)
).
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